Continued fraction and decimal expansions of an irrational number

For an irrational number x and n?1, we denote by kn(x) the exact number of partial quotients in the continued fraction expansion of x given by the first n decimals of x and pn(x)/qn(x) the nth convergent of x. Let

     

Explicit evaluations of Ramanujan-Göllnitz-Gordon continued fraction

In this paper, we establish several theorems for the explicit evaluations of Ramanujan-G?llnitz-Gordon continued fraction by using some parameterizations of Ramanujan’s theta-functions. Nayandeep Deka Baruah, Corresponding author. Research partially supported by grant SR/FTP/MA-02/2002 from DST, Govt. of India. Authors’ address: Nayandeep Deka Baruah and Nipen Saikia, Department of Mathematical Sciences, Tezpur University, Napaam-784028, Assam, India     

Hausdorff dimension of certain sets arising in continued fraction expansions

This paper is concerned with the fractional dimensions of some sets of points with their partial quotients obeying some restrictions in their continued fraction expansions. The Hausdorff dimension of the following set, which shares a dichotomy law according to Borel-Bernstein's theorem, is completely determined

     

Cubic Fields with Index 2

 Let d be a squarefree integer with allowed. If (mod 8) it is shown that there do not exist any cubic fields with index 2 whose splitting field contains . If (mod 8) it is shown that there exist infinitely many cubic fields with index 2 and minimal index 2 whose splitting field contains . (Received 23 May 2001)     

Expressible sets of sequences with Hausdorff dimension zero

We give conditions sufficient for sequences consisting of real numbers to ensure that their expressible sets have Hausdorff dimension zero.     

Metric Theorems for Distribution Measures of Pseudorandom Sequences

 We improve a recent result of Mauduit and Sárk?zy (2000) on the well-distribution measure of pseudorandom sequences and complement another of their results on the correlation measure. Received 12 September 2000; in revised form 1 March 2001     

Construction of pseudorandom binary lattices by using the multiplicative inverse

In an earlier work Hubert and the authors of this paper introduced and studied the notion of pseudorandomness of binary lattices. Later in another paper the authors gave a construction for a large family of “good” binary lattices by using the quadratic characters of finite fields. Here, a further large family of “good” binary lattices is constructed by using finite fields and the notion of multiplicative inverse. Authors’ addresses: Christian Mauduit, Institut de Mathématiques de Luminy, CNRS, UMR 6206, 163 avenue de Luminy, Case 907, F-13288 Marseille Cedex 9, France; András Sárk?zy, Department of Algebra and Number Theory, E?tv?s Loránd University, Pázmány Péter sétány 1/C, H-1117 Budapest, Hungary     

Sur l’ordre de grandeur des quotients partiels du développement en fractions continues régulières

 For any irrational , let denote the regular continued fraction expansion of x and define f, for all z > 0 by and by J. GALAMBOS proved that (μ the Gauss measure)

On Polynomially Normal Lattice Configurations

In this paper we extend Champernowne’s construction of a normal sequence in base b to the case and obtain an explicit construction of the generic point of the shift transformation of the set . We prove that the intersection of the constructed configuration with an arbitrary polynomial curve in the plane is a normal sequence in base b.     

Minkowski’s Second Theorem over a Simple Algebra

We generalize the notion of successive minima, Minkowski’s second theorem and Siegel’s lemma to a free module over a simple algebra whose center is a global field. The author was partly supported by the Grant-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science.     

On Strong Uniform Distribution II

 Let ? be a class of real valued integrable functions on [0,1). We will call a strictly increasing sequence of natural numbers an sequence if for every f in ? we have

Digital Expansions with Respect to Different Bases

We consider the g-ary expansion N=∑ _k b _k (N, g)g ^k of non-negative integers N and prove various results on the distribution and the mean value of the k-th digit b _k (N, g) if g varies in an interval of the form 2≤g≤N ^η. As an application we also consider the average value of the sum-of-digits function s(N, g)=∑ _k b _k (N, g). Received 5 November 2001 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Multidimensional continued fractions and a Minkowski function

The Minkowski question mark function can be characterized as the unique homeomorphism of the real unit interval that conjugates the Farey map with the tent map. We construct an n-dimensional analogue of the Minkowski function as the only homeomorphism of an n-simplex that conjugates the piecewise-fractional map associated to the M?nkemeyer continued fraction algorithm with an appropriate tent map. Author’s address: Department of Mathematics, University of Udine, via delle Scienze 208, 33100 Udine, Italy     

A law of the iterated logarithm for heavy-tailed random vectors

For a random vector belonging to the (generalized) domain of operator semistable attraction of some nonnormal law we prove various variants of Chover's law of the iterated logarithm for the partial sum. Furthermore we also derive some large deviation results necessary for the proof of our main theorems. Received: 30 September 1998 / Revised version: 28 May 1999     

On the -Discrepancy of the Hammersley Point Set

 We give a formula for the -discrepancy of the 2-dimensional Hammersley point set in base 2 for all integers p, . Received 18 May 2001; in revised form 18 December 2001     

Vector-valued Jacobi-like forms

We introduce vector-valued Jacobi-like forms associated to a representation of a discrete subgroup in and establish a correspondence between such vector-valued Jacobi-like forms and sequences of vector-valued modular forms of different weights with respect to ρ. We determine a lifting of vector-valued modular forms to vector-valued Jacobi-like forms as well as a lifting of scalar-valued Jacobi-like forms to vector-valued Jacobi-like forms. We also construct Rankin-Cohen brackets for vector-valued modular forms.     

An Elementary Approach to the Twin Primes Problem

Hardy-Littlewood [4] conjectured an asymptotic formula for the number of prime pairs (twin primes) (p, p+2d) with p+2dy, where d N is fixed and y . Up to now, no one has been able to prove this conjecture, but employing Hardy-Littlewoods circle method, Lavrik [5] showed that in a certain sense this formula holds true for almost-all dy/2.In the present paper, we use a completely different method to prove Lavriks almost-all result. Our method is based on an elementary approach developed by Pan Chengdong [7] to the twin primes problem. By a slight modification of our method, we get a corresponding almost-all result for the binary Goldbach problem. From this, according to [3], we derive Vinogradovs [8] well-known Three-Primes-Theorem.     

On the metric theory of the nearest integer continued fraction expansion

Supposek _n denotes either (n) or (p _n) (n=1,2,...) where the polynomial maps the natural numbers to themselves andp _k denotes thek ^th rationals prime. Also let denote the sequence of convergents to a real numberx and letc _n(x)) _n=1 be the corresponding sequence of partial quotients for the nearest integer continued fraction expansion. Define the sequence of approximation constants _n(x)) _n=1 by

On the <Emphasis Type="Italic">DOTU</Emphasis>-Factorization Problem in Dimension Four

 We prove that any basis of a non-degenerate 4-dimensional lattice with sufficiently small (positive) homogeneous minimum can be represented in the form DOTU. This is of interest in connection with Minkowski’s conjecture about the product of inhomogeneous linear forms. Received 23 September 2001 RID="a" ID="a" Dedicated to Prof. Edmund Hlawka on the occasion of his 85th birthday     

Non-crossed products over function fields

Noncrossed product division algebras are constructed over all function fields and iterated power series fields over global fields, using Hilbert's Irreducibility Theorem and the construction of [B]. Minimum indexes obtained are p ² for odd p and 2³ otherwise. Examples are obtained with large index to exponent ratio. Received: 12 February 2001 / Revised version: 26 November 2001